<i>Topology and Geometry for Physicists</i>

Nash, Charles; Sen, Siddhartha; Stachel, John

doi:10.1119/1.14570

Cited by 284 publications

(461 citation statements)

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“…We refer the reader to the literature [25,36,80,81,83,84] for more details. We note that nothing depended critically on the fact that we w ere considering a non-contractible loop, it could well have been any non-contractible compact manifold, for example a sphere S 2 .…”

Section: Search For a Sphaleron In The Skyrme Model: Morse Theorymentioning

confidence: 99%

“…W e will follow the same convention as Manton and use indices i; j; k; l with reference to the coordinate basis and m; n; o; p with reference to the orthonormal basis. If the reader has di culty with these notions, we recommend the references already mentioned [23,24,25,26].…”

Section: Geometrical Framework For the Skyrme Modelmentioning

confidence: 99%

“…Following the convention used by Manton [28], we will use the indices (25) where summation over m is implicit.X then has coordinates X m in this coordinate system.…”

Section: Non-linear Deformation Of a Bodymentioning

confidence: 99%

“…Manton, among others (our original contributions to this subject are secondary). Since these advances generally use the language and formalism of di erential geometry, the reader should be familiar with these notions (any standard course on tensor analysis/di erential geometry/general relativity should be adequate [23,24,25,26]). …”

Section: Introductionmentioning

confidence: 99%

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Recent mathematical developments in the Skyrme model

Gisiger

1998

Physics Reports

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In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the de nition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy con guration in any sector of xed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B = 2 . W e then describe the work done on the multibaryon minima using rational maps, on the topology of the con guration space and the possible implications of Morse theory. Next we turn to recent w ork on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, speci cally the gradient o w method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering.

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Section: Search For a Sphaleron In The Skyrme Model: Morse Theorymentioning

confidence: 99%

Section: Geometrical Framework For the Skyrme Modelmentioning

confidence: 99%

“…Following the convention used by Manton [28], we will use the indices (25) where summation over m is implicit.X then has coordinates X m in this coordinate system.…”

Section: Non-linear Deformation Of a Bodymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recent mathematical developments in the Skyrme model

Gisiger

1998

Physics Reports

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“…where B M (G) is the set of isomorphism classes of G-bundles over M 16 . By explicit construction we shall prove that, up to isomorphism, the unique G-bundle over ∨ n S 1 is the product bundle ∨ n S 1 × G (which is a purely topological result).…”

Section: Proof Of the Theoremmentioning

confidence: 99%

Geometry of the Aharonov–Bohm Effect

2007

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We show that the connection responsible for any abelian or non abelian Aharonov-Bohm effect with n parallel "magnetic" flux lines in R 3 , lies in a trivial G-principal bundle P → M , i.e. P is isomorphic to the product M × G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering spaceM → M , where path integrals are computed, and the associated bundle P × G C m → M , where the wave function and its covariant derivative are sections.Key words: Aharonov-Bohm effect; fibre bundle theory; gauge invariance.As is well known, the magnetic Aharonov-Bohm (A-B) effect 1,2 is a gauge invariant, non local quantum phenomenon, with gauge group U (1), which takes place in a non simply connected space. It involves a magnetic field in a region where an electrically charged particle obeying the Schroedinger equation cannot enter, i.e. the ordinary 3-dimensional space minus the space occupied by the solenoid producing the field; in the ideal mathematical limit, the solenoid is replaced by a flux line. Locally, the particle couples to the magnetic potential A but not to the magnetic field B; however, the effect is gauge invariant since it only depends on the flux of B inside the solenoid.

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O(3) Electrodynamics

Evans¹

Modern Nonlinear Optics, Part II

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Topology and Geometry for Physicists

Cited by 284 publications

References 0 publications

Recent mathematical developments in the Skyrme model

Recent mathematical developments in the Skyrme model

Geometry of the Aharonov–Bohm Effect

O(3) Electrodynamics

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